Theorem inres

Description: Move intersection into class restriction. (Contributed by NM, 18-Dec-2008)

	Ref	Expression
Assertion	inres	|- ( A i^i ( B |` C ) ) = ( ( A i^i B ) |` C )

Proof

Step	Hyp	Ref	Expression
1		inass	|- ( ( A i^i B ) i^i ( C X. _V ) ) = ( A i^i ( B i^i ( C X. _V ) ) )
2		df-res	|- ( ( A i^i B ) |` C ) = ( ( A i^i B ) i^i ( C X. _V ) )
3		df-res	|- ( B |` C ) = ( B i^i ( C X. _V ) )
4	3	ineq2i	|- ( A i^i ( B |` C ) ) = ( A i^i ( B i^i ( C X. _V ) ) )
5	1 2 4	3eqtr4ri	|- ( A i^i ( B |` C ) ) = ( ( A i^i B ) |` C )